Sets of Fractional Dimensions (IV): On Rational Approximation to Real Numbers

Besicovitch, A. S.

doi:10.1112/jlms/s1-9.2.126

Cited by 185 publications

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“…Three applications of the main theorem will be given, one of which includes the case when ip is not necessarily a descreasing function. The set W(m,n;\fi) has been studied extensively; results about its Lebesgue measure can be found in [16,10]; about its Hausdorff dimension in [14,15,1,8,2,7,5]; and about its Hausdorff measure in [15,4].Define the unique number 0 £ v(Q) < in such that …”

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confidence: 99%

A note on the theorem of Jarník-Besicovitch

Dickinson¹

1997

Glasgow Math. J.

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This note draws together and extends two recent results on Diophantine approximation and Hausdorff dimension. The first, by Hinokuma and Shiga [12], considers the oscillating error function |sin^|/^T rather than the strictly decreasing function q~T of Jarnik's theorem. The second is Rynne's extension [17] to systems of linear forms of Borosh and Fraenkel's paper [3] on restricted Diophantine approximation with real numbers. Rynne's result will be extended to a class of general error functions and applied to obtain a more general form of [12] in which the error function is any positive function.Before stating the problem some notation and definitions are introduced. Define ||v|| to be the distance of the nearest integer vector p e Z" from v e W, that is ||v|| = |v -p^. Let X be an m X n real matrix. Then is a system of n real linear forms in m variables. The set of ip well-approximable mXn matrices is defined as W(m,n; «/0 = {X e U mn : \\qX\\ < «K|q|) for infinitely many q e Z m }.This paper is concerned with the Hausdorff dimension of W(m,n;ijj) when instead of running over all q in Z m , we look at the q confined to a subset Q <=, Z m . Three applications of the main theorem will be given, one of which includes the case when ip is not necessarily a descreasing function. The set W(m,n;\fi) has been studied extensively; results about its Lebesgue measure can be found in [16,10]; about its Hausdorff dimension in [14,15,1,8,2,7,5]; and about its Hausdorff measure in [15,4].Define the unique number 0 £ v(Q) < in such that \ qr ]q Te <°° for e>O.J This is the exponent of convergence. From now on W Q (m,n; i/») will be used to denote the set W Q (m, n; i/0 = {X e W"" : ||qAl < i^(|q|) for infinitely many q e Q).Using this definition Rynne obtained the following result.Note that it follows from Groshev's theorem [10] that if r < v(Q)/m then W Q has full Lebesgue measure.As mentioned above this theorem extends the results of Borosh and Fraenkel [3], Glasgow Math.

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confidence: 99%

A note on the theorem of Jarník-Besicovitch

Dickinson¹

1997

Glasgow Math. J.

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“…[17] or [12]. Moreover, Jarník [16] proved in 1929 that dim{ξ ∈ [0, 1] : µ(ξ) ≥ µ} = 2 µ , (1.2) a result established independently by Besicovitch [8] a few years later. In 1931 Jarník [17] refined his result from [16] by using generalized Hausdorff measures.…”

Section: Introductionmentioning

confidence: 80%

Hausdorff Dimension and Diophantine Approximation

Bugeaud

2013

Further Developments in Fractals and Related Fields

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“…By an application of Lemma 1 as above, they contain, respectively, the unions of intervals corresponding to the sets of triples: 3 . They also contain, respectively, the intervals corresponding to the sets of triples:…”

Section: Notation Let a And B Be Real Numbers If {B}mentioning

confidence: 99%